Some Functors on the Category of Sheaves

Brian Shin March 1, 2017

Abstract In this talk, we’d like to study some functors on categories of sheaves that are induced by continuous maps. More precisely, we will see that a map p : Y X of schemes will induce several functors between ! Sh(Yet´ ) and Sh(Xet´ ). We’ll be studying properties of these functors and relationships among them. Most of the results of this talk can be generalized to the context of continuous maps between sites. Everything in this talk comes from I.8 of [1] §

1 Background on the Category of Sheaves

Recall that X is the full subcategory of Sch/X consisting of etale´ morphisms U X. The category X et´ ! et´ comes equipped a Grothendieck topology, the etale´ topology. We can then form the category Pre(Xet´ ) of presheaves of abelian groups, together with the reflective subcategory Sh(Xet´ ) consisting of sheaves of abelian groups. (A subcategory is reflective if the inclusion functor admits a left adjoint.) Finally, recall that the category of (pre)sheaves on a site with values in an abelian category is again an abelian category. This is essentially due to the fact that short exact sequences can be computed stalkwise.

2 Direct Images

Since the pullback of an etale´ morphism is always an etale´ morphism, we can push presheaves through morphisms. More precisely, we have the following definition.

Definition 2.1. Let p : Y X be a morphism of schemes, and let be a presheaf on Y . For any object ! F et´ U X of Xet´ , we can define p (U)= (U X Y). For any morphism U V in Xet´ , we get a morphism ! ⇤F F ⇥ ! U X Y V X Y, which induces a morphism p (V) p (U). This data determines a presheaf p ⇥ ! ⇥ ⇤F ! ⇤F ⇤F on X called the direct image of along p : Y X. et´ F ! Proposition 2.2. If is a sheaf, then so is p . F ⇤F The direct image construction gives us a functor p : Pre(Yet´ ) Pre(Xet´ ). This functor is evidently ⇤ ! exact, since exactness is checked at the level of open sets. The fact that Sh(Yet´ ) is a reflective subcategory implies the following proposition.

Proposition 2.3. The functor p : Sh(Y´et) Sh(X´et) is left exact. ⇤ ! Example 2.4 (Failure of Right Exactness). Consider the a k-scheme a : X Spec k for some algebraically ! closed field k. The functor Sh((Spec k) ) Ab sending to (Spec k) is an equivalence of categories, et´ ! F F and under this equivalence, the direct image functor a : Sh(Xet´ ) Sh((Spec k)et´ ) is naturally isomorphic to the global sections functor, which is not generally right⇤ exact. ! Example 2.5 (Skyscraper Sheaf). Let i : x X be a geometric point. We again have that Sh(x ) is equivalent ! et´ to the category of abelian groups. Under this equivalence, the direct image i G of an abelian group G is then the skyscraper sheaf Gx at x. ⇤

1 If we have a geometric point i : y Y and a morphism p : Y X, then we get a geometric point ! ! x = p(y)=p i : y X. If we have a sheaf on Yet´ , then the universal property of colimits induces a canonical map ! F (p )x y. ⇤F !F In general, this map is neither injective nor surjective. The following proposition sheds some light on this situation. Proposition 2.6. (a) Let p : V X be an open immersion and let be a sheaf on V . Then ! F ´et x if x V (p )x = F 2 ⇤F ? if x V. ⇢ 62 (b) Let p : Z X be a closed immersion and let be a sheaf on Z . Then ! F ´et x if x Z (p )x = F 2 ⇤F 0 if x Z. ⇢ 62 (c) Let p : Y X be a finite map and let be a sheaf on Y . Then ! F ´et d(y) (p )x = ⇤ y F y x F M7! where d(y) is the separable degree of k(y) over k(x). In particular, if p is an ´etalemap of degree d between varieties over an algebraically closed field, then

d (p )x = . ⇤F Fx Corollary 2.7. If p : Y X is finite or a closed immersion, then the functor p : Sh(Y´et) Sh(X´et) is exact. ! ⇤ ! The following examples are a theme and variations. The first comes from topology, where sheaves can be pushed forward along continuous maps as one would imagine. Example 2.8. Let X R2 be the open disk of radius 1 centered at the origin, let U = X (0, 0) , and let ⇢ { } i : U X be the inclusion. Let be the locally constant sheaf on U corresponding to a p (U, u)-module ! F 1 F, where u U is some basepoint. (Recall that p1(U, u) = Z, so a p1(U, u)-module is the same thing as a 1 2 ⇠ Z[t, t ]-module.) Then p1(U,u) (i )(0,0) = F , ⇤F the elements of F fixed by p1(U, u). Example 2.9. Let X = A1 for some algebraically closed field k, let U = X 0 , and let i : U X be the k { } ! open immersion. Let be the locally constant sheaf on U corresponding to a p (U, u)-module F, where F 1 u U is some geometric point. (Recall that p (U, u) = Z.) Then ! 1 ⇠ p1(U,u) (i )0 = F b , ⇤F the elements of F fixed by p1(U, u). Example 2.10. Let X = Spec R for some Henselian discrete valuation ring R, let U = Spec K where K is the field of fractions of R, and let i : U X be the inclusion. Let be the locally constant sheaf on U ! F corresponding to a p (U, u)-module F, where u U is the geometric point corresponding to K , Ksep. 1 ! ! (Recall that p (U, u) = Gal(Ksep/K).) Let I p (U, u) be the subgroup of elements acting trivially on the 1 ⇠ ✓ 1 residue field of R. (That is, I is the inertia group.) Then

I (i )x = F , ⇤F the elements of F fixed by p (U, u), where x X is the unique closed point. 1 ! We record here a result about ”functoriality” of the direct image. Proposition 2.11. If we have morphisms f : Z Y and p : Y X of schemes, then we have (p f) = p f . ! ! ⇤ ⇤ ⇤

2 3 Inverse Image

Let p : Y X be a morphism of schemes and let be a presheaf on X . We can define a presheaf on ! F et´ F 0 Y , called the inverse image of , by et´ F 0(V)=lim (U), F F ! where the colimit is taken over the collection of commutative diagrams

V U (1)

Y p X

with U X etale.´ Constructing the inverse image presheaf is functorial, yielding a functor ( ) : Pre(X ) ! 0 et´ ! Pre(Yet´ ). If we fill in the above commutative square with the appropriate pullback diagram, we get

V ! U Y U ⇥X

Y p X

If we have a presheaf on Yet´ , then this second diagram implies that we have natural bijective correspon- dences between the followingG collections:

morphisms , • F 0 !G morphisms (U) (V) for every commutative diagram 1 for all diagrams compatible in a suitable • way, and F !G morphisms p . • F! ⇤G In other words, we have an adjunction ( )0 : Pre(Xet´ ) ⌦ Pre(Yet´ ) : p . ⇤ If we take to be a sheaf on X , then it is not generally true that will be a sheaf on Y . To remedy F et´ F 0 et´ this, we can sheafify in order to get a sheaf p on Y . This yields a functor p : Sh(X ) Sh(Y ). For ⇤F et´ ⇤ et´ ! et´ every sheaf on Y , we have a sequence of natural bijections G et´

HomSh(Y )(p⇤ , ) = HomPre(Y )( 0, ) = HomPre(Y )( , p )=HomSh(Y )( , p ) et´ F G ⇠ et´ F G ⇠ et´ F ⇤G et´ F ⇤G

Thus, the adjunction above then gets upgraded to an adjunction p⇤ : Sh(Xet´ ) ⌦ Sh(Yet´ ) : p . ⇤ Proposition 3.1. Let be a sheaf on X , and let p : U X be an ´etalemorphism. Then we have a natural F ´et ! isomorphism U = p . F| ´et ⇠ ⇤F Proposition 3.2. Let i : x X be a geometric point and let be a sheaf on X . Then (i )(x)= . ! F ´et ⇤F Fx Proposition 3.3. If we have morphisms f : Z Y and p : Y X of schemes, then we have (p f) = f p . ! ! ⇤ ⇤ ⇤ Corollary 3.4. The functor p : Sh(X ) Sh(Y ) is exact. ⇤ ´et ! ´et 4 Extension by Zero

Again, consider a morphism p : Y X of schemes. Since the functor p admits a right adjoint, we expect ! ⇤ p⇤ to be right exact. However, corollary 3.4 tells us that it is also left exact. Perhaps the functor p⇤ also admits a left adjoint? Yes! We will construct one in the special case that p : Y X is an open immersion. !

3 Definition 4.1. Let j : U X be an open immersion. Let be a presheaf on U . For every etale´ map ! F et´ f : V X, define ! (V) if f(V) U (V)= F ✓ F! 0 else ⇢ This, together with the obvious restriction maps, yields a presheaf on X . F! et´ Again, if is a sheaf on U , there is no gaurantee that is a sheaf on X . Again, this is not so bad, F et´ F! et´ since we can sheafify to obtain a sheaf j on X . This gives us a functor j : Sh(U ) Sh(X ). !F et´ ! et´ ! et´ Proposition 4.2. Let j : U X be an open immersion. Then we have an adjunction j : Sh(U ) Sh(X ) : j . ! ! ´et ⌦ ´et ⇤ References

[1] James S. Milne. Lectures on etale cohomology (v2.10), 2008. Available at www.jmilne.org/math/.

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